Google Cloud Run is a fully managed platform that enables deployment and scaling of container-based serverless workloads. It abstracts away infrastructure management, allowing developers to focus on writing and packaging code as container images. Monitoring concerns arise when traffic reaches the service, such as auto-scaling behavior, CPU and memory usage, and resource utilization. Key metrics to monitor include CPU utilization, memory allocation time, request count, request latency, billable instance time, and container instance count. These metrics help ensure that serverless container instances have sufficient resources, are performing optimally, and are utilizing resources efficiently. By monitoring these key metrics, developers can optimize resource allocation, reduce costs, and improve the reliability and responsiveness of their serverless containerized applications on Cloud Run.